
as a vector space, \({^{\mathrm {BL}}\mathcal {H}}(T_= Y)\otimes \mathcal {H}_{W^v,  (we write \theta *h instead of \theta \otimes h for \theta \in Y) and h\in \mathcal {H}_{W^v,),}\item {^{\mathrm {BL}}\mathcal {H}}(T_ is equipped with the unique product * which turns it into an associative algebra and such that, for \theta \in Y) and s\in {S}, we have: T_{s}*\theta -{^s\theta }*T_{s} =\tilde{\Omega }_s(\theta -{^s\theta }), where \tilde{\Omega }_s(\theta )=Q_s^T(\theta -{^s\theta }) and Q_s^T=\frac{(\sigma _s^2-1)+\sigma _s(\sigma _s^{\prime }-\sigma _s^{\prime -1})Z^{-\alpha _s^\vee }}{1-Z^{-2\alpha _s^\vee }}\index {Q_s^T}.}\)
By [1]}, such an algebra exists and is unique.
The Bernstein-Lusztig Hecke algebra and the Iwahori-Hecke algebra
Let \(C^v_f=\lbrace x\in \mathbb {A}|\alpha _i(x)>0\forall i\in I\rbrace \), \(\bigcup _{w\in W^v} w.\overline{C}^v_f\) be the Tits cone and \(Y^+=Y\cap \index {Y^+}.\)
Definition 2.2 
The Bernstein-Lusztig-Hecke algebra of  \((\mathbb {A},(\sigma _s)_{s\in {S}},(\sigma ^{\prime }_s)_{s\in {S}})\)  over \( is the subalgebra \) BLHY,wWvZ̏*Tw=Y,wWvT̏w* Z\( of \) BLH(T\(. The \textbf {Iwahori-Hecke algebra of \) (A,(s)sS,('s)sS)\( over \) } is the subalgebra \(\mathcal {H}_\bigoplus _{\lambda \in Y^+,w\in W^v}Z̏^\lambda *T_w=\bigoplus _{\lambda \in Y^+,w\in W^v}T̏_w*Z^\lambda \)  of \({^{\mathrm {BL}}\mathcal {H}}_. Note that for \) G\( reductive, we recover the usual Iwahori-Hecke algebra of \) G\(, since \)  A\(.\)
Remark 2.3

The algebra \({^{\mathrm {BL}}\mathcal {H}}_ was first defined in \cite [Theorem 6.2]{bardy2016iwahori} without defining \) BLH(T\(. Let \) K\( be a non-Archimedean local field and \) q\( be its residue cardinal. Let \) G\( be the minimal Kac-Moody group associated with \) S=(A,X,Y,(i)iI,(i)iI)\(\index {\mathcal {S}} and \) G=G(K)\( (see \cite [Section 8]{remy2002groupes} or \cite {tits1987uniqueness} for the definition). Take \) s='s=q\( for all \) sS\(. Then \) H is the Iwahori-Hecke algebra of \(G\)  (see [2]}). In the case where \(G\)  is an untwisted affine Kac-Moody group, these algebras were introduced in [3]}. Note also that our frameworks is more general than the one of split Kac-Moody groups over local fields. It enables for example to study the Iwahori-Hecke algebras associated with almost split Kac-Moody groups over local fields, as in [2]}. In this case we do not have necessarily \(\sigma _s=\sigma _s^{\prime }=\sigma _t=\sigma _t^{\prime }\)  for all \(s,t\in {S}\) .

Let \(s\in {S}\) . Then if \(\sigma _s=\sigma ^{\prime }_s\) , \(Q_s^T=\frac{\sigma _s^2-1}{1-Z^{-\alpha _s^\vee }}\) .

 Let \(s\in {S}\) . Then \(\tilde{\Omega }_s(Y])\subset Y]\)  and \(\tilde{\Omega }_s(Y^+])\subset Y^+]\)  Indeed, let \(\lambda \in Y\) . Then \(Q_s^T(Z^\lambda -Z^{s.\lambda })=Q^T_s.Z^\lambda (1-Z^{-\alpha _s(\lambda )\alpha _s^\vee })\) . Assume that \(\sigma _s=\sigma _s^{\prime }\) . Then
\( \frac{1-Z^{-\alpha _s(\lambda )\alpha _s^\vee }}{1-Z^{-\alpha _s^\vee }}=\left\lbrace \begin{aligned} &\sum _{j=0}^{\alpha _s(\lambda )-1}Z^{-j\alpha _s^\vee } &\mathrm {\ if\ }\alpha _s(\lambda )\ge 0\\ & -Z^{\alpha _s^\vee }\sum _{j=0}^{-\alpha _s(\lambda )-1}Z^{j\alpha _s^\vee } &\mathrm {\ if\ }\alpha _s(\lambda )\le 0,\end{aligned}\right.\) 
and thus \(Q_s^T(Z^\lambda -Z^{s.\lambda })\in Y]\) . If \(\sigma _s^{\prime }\ne \sigma _s\) , then \(\alpha _s(Y)=2\mathbb {Z}\)  and a similar computation enables to conclude. In particular, \({^{\mathrm {BL}}\mathcal {H}}_ and \) H are subalgebras of \({^{\mathrm {BL}}\mathcal {H}}(T_\) .

In [1]} and [6]}, we used the basis \((H_w)_{w\in W^v}\)  instead of \((T_{w})_{w\in W^v}\)  in the presentation of \(\mathcal {H}_{W^v, and {^{\mathrm {BL}}\mathcal {H}}_. If w\in W^v, w=s_1\ldots s_k, with k=\ell (w) and s_1,\ldots ,s_k\in {S}, we have T_w=\sigma _{s_1}\ldots \sigma _{s_k} H_w.}\)
Principal series representations
In this subsection, we introduce the principal series representations of \({^{\mathrm {BL}}\mathcal {H}}_.\) We now fix \((\mathbb {A},(\sigma _s)_{s\in {S}},(\sigma ^{\prime }_s)_{s\in {S}})\)  as in Subsection REF . Let \({^{\mathrm {BL}}\mathcal {H}}_ be the Iwahori-Hecke and the Bernstein-Lusztig Hecke algebras of \) (A,(s)sS,('s)sS)\( over \) .
Let \(T_ \mathrm {Hom}_{\mathrm {Gr}}(Y,*)\) be the group of group morphisms from \(Y\)  to \(*\) . Let \(\tau \in T_. Then \)\( induces an algebra morphism \) :Y] by the formula \(\tau (\sum _{\lambda \in Y} a_\lambda Z^\lambda )=\sum _{\lambda \in Y} a_\lambda \tau (\lambda )\) , for \(\sum a_\lambda Z^\lambda \in Y]\) . This equips \( with the structure of a \) Y]\(-module.\)
Let \(I_\tau =\mathrm {Ind}^{{^{\mathrm {BL}}\mathcal {H}}__{Y]}(\tau )={^{\mathrm {BL}}\mathcal {H}}__{Y]} \index {I_\tau }. As a vector space, I_\tau =\bigoplus _{w\in W^v} _\tau , where \mathbf {v}_\tau  is some symbol. The actions of {^{\mathrm {BL}}\mathcal {H}}_ on I_\tau  is as follows. Let h=\sum _{w\in W^v} T_w P_w\in {^{\mathrm {BL}}\mathcal {H}}_, where P_w\in Y] for all w\in W^v. Then h.\mathbf {v}_\tau =\sum _{w\in W^v} \tau (P_w)T_w\mathbf {v}_\tau . In particular, I_\tau  is a principal {^{\mathrm {BL}}\mathcal {H}}_-module generated by \mathbf {v}_\tau .}\) If \(A\)  is a vector space over \( and \) B\( is a set, we denote by \) A(B)\( the set of families \) (ab)bB\( such that \) {bB|ab0}\( is finite. We regard the elements of \) Y]\( as polynomial functions on \) T by setting:
\(\tau (\sum _{\lambda \in Y}a_\lambda Z^\lambda )=\sum _{\lambda \in Y}a_\lambda \tau (\lambda ),\) 
for all \((a_\lambda )\in {(Y)}\) . The ring \(Y]\)  is a unique factorization domain. Let \(\theta \in Y)\)  and \((f,g)\in Y]\times Y]^*\)  be such that \(\theta =\frac{f}{g}\)  and \(f\)  and \(g\)  are coprime. Set \(\mathcal {D}(\theta )=\lbrace \tau \in T_\tau (g)\ne 0\rbrace \) . Then we regard \(\theta \)  as a map from \(\mathcal {D}(\theta )\)  to \( by setting \) ()=f()g()\( for all \) D()\(.\)
If \(\tau \in T_, let \) Y)={fg|f,gY] and  g()0}Y)\(\index {Y)_\tau }. Let \) BLH(T=wWv Tw Y)BLH(T\(\index {{^{\mathrm {BL}}\mathcal {H}}(T__\tau ,{^{\mathrm {BL}}\mathcal {H}}(T__\tau }. This is a not a subalgebra of \) BLH(T\( (consider for example \) 1Z-1*Ts=Ts*1Zs.-1+...\( for some well chosen \) Y\(, \) sS\( and \) T). It is however an \({\mathcal {\mathcal {H}}_{W^v,\mathcal {F}}}-Y)_\tau \)  bimodule. For \(\tau \in T_, we define \) ev:BLH(THWv,F\(\index {ev_\tau } by \) ev(h)=h()=wWv Tww()\( if \) h=wWvTw wBLH(T\(. This is a morphism of \) HWv,F-Y)\(-bimodules.\)

Decomposition of \(W_\tau \)
For \(r=wsw^{-1}\in {R}\) , with \(w\in W^v\)  and \(s\in {S}\) , we define \(Q^T_r={^w\left(Q_s^T\right)}\)  and \(\sigma _r=\sigma _s\) . This does not depend on the choices of \(w\)  and \(s\) . For \(r\in {R}\) , set \(\zeta _r=-Q^T_r+\sigma _r^2\) . Write \(\zeta _r=\frac{\zeta _r^{\mathrm {num}}}{\zeta _r^{\mathrm {den}}}\), with \(\zeta _r^{\mathrm {num}},\zeta _{r}^{\mathrm {den}}\)  are coprime elements of \(Y]\) .
For \(\tau \in T_, set \) W={wWv| w.=}\(\index {W_\tau }, \) ()={| den()=0}\(\index {\Phi ^\vee _{(\tau )}}, \) (),+=()+\(, \) R()={r=rR|()}\(\index {{R}_{(\tau )}} and \)\(W_{(\tau )}=\langle {R}_{(\tau )}\rangle =\langle \lbrace r=r_{\alpha ^\vee }\in {R}| \zeta _{\alpha ^\vee }^{\mathrm {den}}(\tau )=0\rbrace \rangle \subset W^v.\)\(\index {W_{(\tau )}}\) By [1]}, \(W_{(\tau )}\subset W_\tau \) . When \(\alpha _s(Y)=\mathbb {Z}\)  for all \(s\in {S}\) , then \(W_{(\tau )}=\langle W_\tau \cap {R}\rangle \) . Let
\(R_\tau =\lbrace w\in W_\tau |w.\Phi ^\vee _{(\tau ),+}=\Phi ^\vee _{(\tau ),+}\rbrace .\) 
By [6]}, \(W_{(\tau )}\)  is normal in \(W_\tau \)  and \(w_\tau =R_\tau \ltimes W_{(\tau )}\) . We defined in [1]} a set \({S}_\tau \subset {R}\cap W_{(\tau )}\)  for which \((W_{(\tau )},{S}_\tau )\)  is a Coxeter system. We denote by \(\le _\tau \)  the Bruhat order and by \(\ell _\tau \)  the length on \((W_{(\tau )},{S}_\tau )\) . Then by [1]}, for all \(w,w^{\prime }\in W_{(\tau )}\) , we have \(w\le _\tau w^{\prime }\)  implies \(w\le w^{\prime }\) .

Weights and intertwining operators
If \(\tau \in T_, we denote by \) End(I)\( the algebra of endomorphisms of \) BLH-modules of \(I_\tau \) . Let
\(I_\tau (\tau )=\lbrace x\in I_\tau |\theta .x=\tau (\theta ).x\forall \theta \in Y]\rbrace \) 
and
\(I_\tau (\tau ,\mathrm {gen})=\lbrace x\in I_\tau |\forall \theta \in Y], (\theta -\tau (\theta ))^n.x=0,\ \forall n\gg 0\rbrace \supset I_\tau (\tau ).\) 
For \(s\in {S}\) , one sets
\(F_s=T_s+Q_s^T\in {^{\mathrm {BL}}\mathcal {H}}(T_\index {F_s}.\) 
Let \(x\in I_\tau (\tau )\) . Define \(\Upsilon _x\in \mathrm {End}(I_\tau )\)  by \(\Upsilon (h.\mathbf {v}_\tau )=h.x\) , for \(h\in {^{\mathrm {BL}}\mathcal {H}}_. Then it is easy to check that \) :I()End(I)\( is well defined and is an isomorphism of vector spaces.\)
Let \(w\in W^v\) . Let \(w=s_1\ldots s_r\)  be a reduced expression of \(w\) , with \(k=\ell (w)\)  and \(s_1,\ldots ,s_k\in {S}\) . Set
\(F_w=F_{s_r}\ldots F_{s_1}=(T_{s_r}+Q_{s_r}^T)\ldots (T_{s_1}+Q_{s_1}^T)\in {^{\mathrm {BL}}\mathcal {H}}(T_.\) 

Lemma 2.4 (see [1]}) Let \(w\in W^v\) .

 The element \(F_w\in {^{\mathrm {BL}}\mathcal {H}}(T_\)  is well defined, i.e it does not depend on the choice of a reduced expression for \(w\) .

 \(F_w-T_w\in {^{\mathrm {BL}}\mathcal {H}}(T_^{<w}=\bigoplus _{v<w}T_vY)\) .

 If \(\theta \in Y)\) , then \(\theta *F_w=F_w*{^{w^{-1}}}\theta \) .

 If \(\tau \in T_ is such that \) FwBLH(T\(, then \) .Fw().v=(w.()) Fw().v\(, for every \) Y]\(.\)


We set
\(\mathcal {U}_\lbrace \tau \in T_\tau (\zeta _{r}^{\mathrm {num}})\ne 0\forall r\in {R}\rbrace .\) 
When \(\sigma _s=\sigma _s^{\prime }=\sqrt{q}\) , for \(q\in \mathbb {R}_{>0}\) , \(\mathcal {U}_\lbrace \tau \in T_\tau (\alpha ^\vee )\ne q,\forall \alpha ^\vee \in \Phi ^\vee \rbrace \) .
By [6]}, if \(w_R\in R_\tau \) , then \(F_{w_R}\in {^{\mathrm {BL}}\mathcal {H}}(T__\tau \)  and \(F_{w_R}(\tau ).\mathbf {v}_\tau \in I_\tau (\tau )\) . Let \(\psi _{w_R}=\Upsilon _{F_{w_R}(\tau ).\mathbf {v}_\tau }\in \mathrm {End}(I_\tau )\) .
Set
\(I_\tau (\tau ,\mathrm {gen},W_{(\tau )})= \left({^{\mathrm {BL}}\mathcal {H}}_\bigoplus _{w\in W_{(\tau )}} F_w *Y)\right).\mathbf {v}_\tau .\) 
which is well defined by [1]}.
\( \sum _{k\in \mathbb {N},s_1,\ldots , s_k\in {S}_\tau } _\tau (\mathcal {K}_{s_1}*\ldots *\mathcal {K}_{s_k}).\mathbf {v}_\tau ,\) 
By [6]}, if \(\tau \in \mathcal {U}_, we have \begin{equation}I_\tau (\tau ,\mathrm {gen})=\bigoplus _{w_R\in R_\tau } \psi _{w_R}\big (I_\tau (\tau ,\mathrm {gen},W_{(\tau )})\big ).\end{equation}\)

Description of \(I_\tau (\tau ,\mathrm {gen},W_{(\tau )})\)
In this paper, we prove that if \(\tau \in \mathcal {U}_, then \) I(,W()):=I(,gen,W())I()=\( (see Theorem~\ref {thmWeight_space}) and we deduce information on the submodules of \) I\(. To that end,we begin by describing \) I(,gen,W())\(. For \) sS\(, let \begin{equation}\mathcal {K}_s=F_s-\zeta _s=F_s+Q^T_s-\sigma _s^2\in {^{\mathrm {BL}}\mathcal {H}}(T_.\end{equation} In \cite {hebert2021decompositions} and \cite {hebert2021decompositions}, \) I(,gen,W())\( is described in terms of the \) ev(Ks1*...*Ksk).v\(, where \) s1,...,skS\(. However, if \) wW()\( and \) w=s1...sk=s1'...sk'\( is a reduced expression of \) w\(, with \) k=(w)\( and \) s1,...,sk,s1',...,sk'S\(, we might have \) Ks1...KskKs1'...Ksk'\(. We thus slightly modify the \) Ks\( and define \) Ks\(, for \) sS\(, so that we have \) Ks1...Ksk= Ks1'...Ksk'\(. This enables us to define \) Kw\(, for \) wW()\(. Let \) K=wW() Kw *Y)\(. We prove that \) K\( has a presentation very close to the Bernstein-Lusztig presentation of \) BLH.
In order to study \(I_\tau (\tau ,\mathrm {gen},W_{(\tau )})\) , it is then convenient to describe it as a \({K}_\tau \) -module. However, \({K}_\tau \)  is not contained in \({^{\mathrm {BL}}\mathcal {H}}_. We thus extend the action of \) KBLH on \(I_\tau (\tau ,\mathrm {gen},W_{(\tau )})\)  to an action of \({K}_\tau \)  on \(I_\tau (\tau ,\mathrm {gen},W_{(\tau )})\) .
In subsection REF , we study \({K}_\tau \)  and in subsection REF , we define an action of \({K}_\tau \)  on \(I_\tau (\tau ,\mathrm {gen},W_{(\tau )})\) . In subsection REF , we prove that there can exist \(\tau \in \mathcal {U}_ for which \) S\( is infinite.\)
Bernstein-Lusztig-Hecke structure on \({K}_\tau \)
For \(s\in {S}_\tau \) , we set
\(\tilde{\mathcal {K}}_s=\mathcal {K}_s+\sigma _s^2=F_s+Q_s^T\in {^{\mathrm {BL}}\mathcal {H}}(T_\) 
 Note that if \(\tau (\lambda )=1\)  for all \(\lambda \in Y\) , we have \({S}_\tau ={S}\)  and
\(\tilde{\mathcal {K}}_s=T_s\) 
for \(s\in {S}\) .
Let \(s\in {S}_\tau \) . For \(\theta \in Y)\) , set \(\tilde{\Omega }_s(\theta )=Q_s^T(\theta -{^s\theta })\) . Then similarly as in Remark REF  (REF ) and [1]} we have
\(\tilde{\Omega }_s(Y])\subset Y]\text{ and }\tilde{\Omega }_s\big (Y)_\tau \big )\subset Y)_\tau .\) 
The aim of this subsection is to prove the following proposition.
Proposition 3.1 
Let \(\tau \in T_.\begin{enumerate}\item Let w\in W_{(\tau )} and w=s_1\ldots s_k be a reduced decomposition of w, with k=\ell _\tau (w) and s_1\ldots ,s_k\in {S}_\tau . Then \tilde{\mathcal {K}}_{w}=\tilde{\mathcal {K}}_{s_1}*\ldots *\tilde{\mathcal {K}}_{s_k}\in {^{\mathrm {BL}}\mathcal {H}}(T__\tau \index {\tilde{\mathcal {K}}_w} is well defined, independently of the choice of the reduced decomposition.\end{enumerate}\item  Let \) sS\( and \) wW()\(. Then \) Ks*Kw={
\( & \tilde{\mathcal {K}}_{sw} &\mathrm {\ if\ }\ell _\tau (sw)=\ell (w)+1\\ & (\sigma _{s}^2-1) \tilde{\mathcal {K}}_{w}+\sigma _s^2 \tilde{\mathcal {K}}_{s w} &\mathrm {\ if\ }\ell _\tau (sw)=\ell _\tau (w)-1 .\) 
.  \(\)
 For \(\theta \in Y)\)  and \(s\in {S}_\tau \) , we have
\(\theta *\tilde{\mathcal {K}}_s=\tilde{\mathcal {K}}_s*{^s\theta }+\tilde{\Omega }_s(\theta ).\) 

The matrix \(\big (\alpha _s(\alpha _r^\vee )\big )_{r,s\in {S}_\tau }\)  is a (possibly infinite) Kac-Moody matrix.

 If \(\tau \in \mathcal {U}_, the space \)\({K}_\tau =\sum _{k\in \mathbb {N}, s_1\ldots ,s_k\in {S}_\tau } \mathcal {K}_{s_1}*\ldots *\mathcal {K}_{s_k}*Y)_\tau =\bigoplus _{w\in W_{(\tau )}}\tilde{\mathcal {K}}_w*Y)_\tau \)\( is a subalgebra of \) BLH(T\( contained in \) BLH(T\(.\)
In particular, \(\bigoplus _{w\in W_{(\tau )}} \tilde{\mathcal {K}}_w*Y)_\tau \)  is almost a Bernstein-Lusztig-Hecke algebra as defined in subsection REF . Note however that \({S}_\tau \)  can be infinite (and thus the \((\alpha _s)_{s\in {S}_\tau }\)  and \((\alpha _{s}^\vee )_{s\in {S}_\tau }\)  are not free), see Lemma REF . The proof of this proposition is a consequence of the lemmas of this subsection below.
Lemma 3.2 Let \(s\in {S}_\tau \) . Then:

\(\mathcal {K}_s^2=-(1+\sigma _s^2)\mathcal {K}_s\) ,

\(\tilde{\mathcal {K}}_s^2=(\sigma _s^2-1)\tilde{\mathcal {K}}_s+\sigma _s^2\) .


By Lemma REF , () and [1]}, we have
\(\begin{aligned}\mathcal {K}_s^2 &=(F_s-\zeta _s)(F_s-\zeta _s)\\&=F_s^2-F_s(\zeta _s+{^s\zeta _s})+\zeta _s.{^s\zeta _s}\\&=\zeta _s.({^s\zeta _s})-F_s(\zeta _s+{^s\zeta _s})+\zeta _s.{{^s\zeta _s}}\\&=(-F_s+\zeta _s)(\zeta _s+{^s\zeta _s})=-(1+\sigma _s^2)\mathcal {K}_s.\end{aligned}\) 
Lemma 3.3 
Let \(\theta \in Y)\)  and \(s\in {S}_\tau \) . Then
\(\theta *\tilde{\mathcal {K}}_s=\tilde{\mathcal {K}}_s*{^s\theta }+\tilde{\Omega }_s(\theta ).\) 

By Lemma REF , we have
\(\begin{aligned}\theta *\tilde{\mathcal {K}}_s&=\theta *\left(F_s+Q_s^T\right)\\&=F_s* {^s\theta }+\theta *Q_s^T \\&=\left(F_s+Q_s^T\right){^s\theta }+Q_s^T(\theta -{^s\theta })\\&= \tilde{\mathcal {K}}_s *{^s\theta } +\tilde{\Omega }_s(\theta ).\end{aligned}\) 
Following [17]}, we define \(\le _\tau \)  on \(\Phi _{(\tau ),+}^\vee \)  as follows. If \(\alpha ^\vee ,\beta ^\vee \in \Phi ^\vee _{(\tau ),+}\) , we write \(\alpha ^\vee \le _{\tau } \beta ^\vee \)  if there exist \(k\in \mathbb {N}\) , \(a\in \mathbb {R}^*_+\) , \(\beta _1^\vee ,\ldots ,\beta _k^\vee \in \Phi _{(\tau ),+}^\vee \)  and \(a_1,\ldots ,a_k\in \mathbb {R}_+\)  such that \(\alpha ^\vee = a\beta ^\vee +\sum _{i=1}^k a_i \beta _i^\vee \) .
Let \(s\in W_{(\tau )}\cap {R}\) . Then by [1]}, we have
\(s\in {S}_\tau \text{ if and only if }s.\big (\Phi _{(\tau ),+}^\vee \setminus \lbrace \alpha _s^\vee \rbrace \big )=\Phi _{(\tau ),+}^\vee \setminus \lbrace \alpha _s^\vee \rbrace .\) 
Lemma 3.4 
Let
\(\Sigma _\tau =\lbrace \alpha ^\vee \in \Phi _{(\tau ),+}^\vee |\forall \beta ^\vee \in \Phi _{(\tau ),+}^\vee , \beta ^\vee \le _\tau \alpha ^\vee \Rightarrow \beta ^\vee =\alpha ^\vee \rbrace .\) 
Then the map \(s\mapsto \alpha _s^\vee \)  from \({S}_\tau \)  to \(\Sigma _\tau \)  is well defined and is a bijection. Moreover, we have
\(\Phi ^\vee _{(\tau ),+}\subset \sum _{\alpha ^\vee \in \Sigma _\tau }\mathbb {N}\alpha ^\vee .\) 

This follows [17]}.
Let \(\alpha ^\vee \in \Sigma _\tau \)  (if such an element exists). Let \(s=r_{\alpha ^\vee }\in {R}_{(\tau )}\)  and \(\beta ^\vee \in \Phi ^\vee _{(\tau ),+}\) . We assume that \(\gamma ^\vee :=-s.\beta ^\vee \in \Phi ^\vee _{(\tau ),+}\) . Then \(\gamma ^\vee =-\beta ^\vee +\alpha _s(\beta ^\vee )\alpha _s^\vee \) . As \(\beta ^\vee \in \Phi ^\vee _+\) , we necessarily have \(\alpha _s(\beta ^\vee )\in \mathbb {N}^*\) . Therefore \(\alpha _s^\vee =\frac{1}{\alpha _s(\beta ^\vee )}(\beta ^\vee +\gamma ^\vee )\)  and \(\beta ^\vee \le _\tau \alpha _s^\vee \) . By definition of \(\Sigma _\tau \) , we deduce \(\beta =\alpha _s^\vee \) . By (REF ) we deduce that \(s\in {S}_\tau \) .
Conversely, let \(s\in {S}_{\tau }\) . Let \(\beta ^\vee \in \Phi ^\vee _{(\tau ),+}\)  be such that \(\beta ^\vee \le _\tau \alpha _s^\vee \) . By definition, we can write \(\alpha _s^\vee = a\beta ^\vee +\sum _{i=1}^k a_i \beta _i^\vee \) , where \(k\in \mathbb {N}\) , \(a\in \mathbb {R}^*_+\) , \((a_i)\in (\mathbb {R}_+)^k\)  and \((\beta _i^\vee )\in \big (\Phi _{(\tau ),+}\big )^k\) . Then
\(s.\beta ^\vee =-\frac{1}{a}(\alpha _s^\vee +\sum _{i=1}^k a_is.\beta _i^\vee )\in \Phi ^\vee _-\cap \Phi ^\vee _{(\tau )}.\) 
By (REF ), we deduce that \(\beta ^\vee =\alpha _s^\vee \)  and thus \(\alpha _s^\vee \in \Sigma _\tau \) . We proved the first part of the lemma. Then (REF ) can be proved as [17]}, using the fact that \(\alpha (\beta ^\vee )\in \mathbb {Z}\) , for all \(\alpha ^\vee ,\beta ^\vee \in \Phi _{(\tau )}^\vee \) .
Lemma 3.5 
The matrix \(\big (\alpha _s(\alpha _r^\vee )\big )_{r,s\in {S}_\tau }\)  is a Kac-Moody matrix.

Let \(r,s\in {S}_\tau \) . Suppose \(\alpha _r(\alpha _s^\vee )\ne 0\) . We have \(r.\alpha _s^\vee =\alpha _s^\vee -\alpha _r(\alpha _s^\vee )\alpha _r^\vee \) . By Lemma REF , \(r.\alpha _s^\vee \lnot \le _\tau \alpha _r^\vee \) , which implies that \(\alpha _r(\alpha _s^\vee )\le 0\) .
Suppose \(\alpha _r(\alpha _s^\vee )=0\) . Then by the case above, \(\alpha _s(\alpha _r^\vee )\le 0\) . Suppose that \(\alpha _s(\alpha _r^\vee )<0\) . Then
\(\alpha _r^\vee =-rs.\alpha _r^\vee -\alpha _s(\alpha _r^\vee )\alpha _s^\vee \text{ and }\alpha _s^\vee =\frac{-1}{\alpha _s(\alpha _r^\vee )}(rs.\alpha _r^\vee +\alpha _r^\vee ).\) 
If \(rs.\alpha _r^\vee \in \Phi ^\vee _+\) , we deduce that \(\alpha _s^\vee \le _{\tau } \alpha _r^\vee \)  and if \(rs.\alpha _r^\vee \) , we deduce that \(\alpha _r^\vee \le _\tau \alpha _s^\vee \) . Since \(\alpha _r^\vee ,\alpha _s^\vee \in \Sigma _\tau \) , this implies in both cases that \(\alpha _r^\vee =\alpha _s^\vee \) : we reach a contradiction. Therefore \(\alpha _s(\alpha _r^\vee )=0\)  and the lemma follows.
If \(a,b\)  are two elements of a ring and \(m\in \mathbb {N}\) , we denote by \((a.b)^{*m}\)  the product \(a.b.a.\ldots \)  with \(m\)  factors.
Lemma 3.6 Let \(w\in W_{(\tau )}\)  and \(w=r_1\ldots r_k= s_1\ldots s_k\)  be two reduced writings of \(w\) , with \(k=\ell _\tau (w)\)  and \(r_1,\ldots , r_k, s_1,\ldots ,s_k\in {S}_\tau \) . Then
\(\tilde{\mathcal {K}}_{r_1}\ldots \tilde{\mathcal {K}}_{r_k}=\tilde{\mathcal {K}}_{s_1}\ldots \tilde{\mathcal {K}}_{s_k}.\) 

Let \(r,s\in {S}_\tau \) . Suppose that the order \(m\)  of \(rs\)  is finite. Let \(h=(\tilde{\mathcal {K}}_r *\tilde{\mathcal {K}}_s)^{*m}-(\tilde{\mathcal {K}}_s*\tilde{\mathcal {K}}_r)^{*m}\in {^{\mathrm {BL}}\mathcal {H}}(T_\) . We want to prove that \(h=0\) . Set \(w_0(r,s)=(rs)^{*m}=(sr)^{*m}\) . Using Lemma REF , (REF ) and [1]}, there exists a family
\((P_u)_{u\in \langle r,s\rangle }\in \left(\mathbb {Z}[\sigma _r,\sigma _s][x_{s,u},x_{r,u^{\prime }}|u,u^{\prime }\in \langle r,s\rangle ]\right)^{\langle r,s\rangle }\) 
such that
\(h=\sum _{u\in \langle r,s\rangle } F_u*P_u\left(\left({^v Q^T_s},^w Q^T_r \right)_{v,w\in \langle r,s\rangle }\right),\) 
where the \(x_{s,u},x_{r,u}\)  are indeterminates.
Let \(A_{r,s}=\begin{pmatrix}2 & \alpha _s(\alpha _r^\vee )\\ \alpha _r(\alpha _s^\vee ) & 2\end{pmatrix}\) . Then \(A_{r,s}\)  is a Kac-Moody matrix by Lemma REF  (it is actually a Cartan matrix since \(\langle r,s\rangle \)  is finite). Let \(X^{\prime },Y^{\prime },\alpha _r^{\prime },\alpha _s^{\prime },\alpha _r^\vee ,\alpha _s^\vee \)  be copies of \(X,Y,\alpha _r,\alpha _s,\alpha _r^\vee ,\alpha _s^\vee \) . Then \(\mathcal {S}^{\prime }=(A_{r,s},X^{\prime },Y^{\prime },(\alpha _r^{\prime },\alpha _s^{\prime }),(\alpha _r^{\prime \vee },\alpha _s^{\prime \vee })\big )\)  is a root generating system.
Let \(W^{\prime }\)  be the associated Weyl group. We add a prime to all the objects previously defined when they refer to the root generating system \(\mathcal {S}^{\prime }\) . Let \(\tau ^{\prime }\in T_\)  be defined by \(\tau ^{\prime }(\lambda )=1\)  for all \(\lambda \in Y^{\prime }\) . Then we have \({S}^{\prime }_{\tau ^{\prime }}=\lbrace r,s\rbrace \subset W^{\prime }\) . By (REF ), \(\tilde{\mathcal {K}}_r^{\prime }=T_r^{\prime }\in {^{\mathrm {BL}}\mathcal {H}}(T_^{\prime }\)  and \(\tilde{\mathcal {K}}_s^{\prime }=T_s^{\prime }\in {^{\mathrm {BL}}\mathcal {H}}(T_^{\prime }\) . Moreover \((T_r^{\prime }*T_s^{\prime })^{*m} -(T_s^{\prime }*T_r^{\prime })^{*m}=T_{w_0^{\prime }}^{\prime }-T_{w_0^{\prime }}^{\prime }=0\) , where \(w_0^{\prime }=(rs)^{*m}=(sr)^{*m}\in W^{\prime }\)  is the element of maximal length of \(W^{\prime }\) . Therefore if \(u\in W^{\prime }\) , we have
\(P_u\big (({^v{Q_r^{\prime T}}},{^w Q_s^{\prime T})_{v,w\in W^{\prime }}}\big )=0.\) 
Let \(Q^{\prime \vee }=\mathbb {Z}\alpha _r^{\prime \vee }\oplus \mathbb {Z}\alpha _s^{\prime \vee }\)  and \(\iota :Q^{\prime \vee }\rightarrow Q^\vee \)  be the \(\mathbb {Z}\) -modules morphism defined by \(\iota (\alpha _r^{\prime \vee })=\alpha _r^\vee \)  and \(\iota (\alpha _s^{\prime \vee })=\alpha _s^\vee \) . Extend \(\iota \)  to a \(-algebra morphism \) :Q']Q]\(. Then \)\( is compatible with the actions of \) W'\( and \) r,s\(. More precisely, if \) :W'r,s\( is the group morphism defined by \) (r)=r\(, \) (s)=s\(, we have \) (w Q'Tt)=(w) QTt,\( for \) wW'\( and \) t{r,s}\(.Therefore \)\(P_u\left(\left({\iota \left(^vQ^{\prime T}_r\right)}, \iota \left({^w Q^{\prime T}_s}\right)\right)_{v,w\in W^{\prime }}\right)=P_u\left(\left({^vQ^T_r},{^w Q^T_s}\right)_{v,w\in \langle r,s\rangle }\right)=0,\)\(for \) ur,s\( which proves that \) h=0\( and hence \) (Kr*Ks)*m=(Ks*Kr)*m\(. By the word property (\cite [Theorem 3.3.1]{bjorner2005combinatorics}), we deduce the lemma.\)
To conclude the proof of Proposition REF , it remains to prove (5). By (2) and (3), \( {K}_\tau =\sum _{k\in \mathbb {N}, s_1,\ldots ,s_k\in {S}_\tau }\tilde{\mathcal {K}}_{s_1}*\ldots *\tilde{\mathcal {K}}_{s_k}*Y)_\tau \)  is a subalgebra of \({^{\mathrm {BL}}\mathcal {H}}(T_\) . Let \(w\in W_{(\tau )}\) . Let \(w=s_1\ldots s_k\)  be a reduced writing of \(w\) , with \(k\in \mathbb {N}\)  and \(s_1,\ldots ,s_k\in {S}_\tau \) . Then by Lemma REF ,
\(\tilde{\mathcal {K}}_{s_1}*\ldots *\tilde{\mathcal {K}}_{s_k}=(F_{s_1}+Q_{s_1}^T)\ldots (F_{s_k}+Q_{s_k}^T)= F_{w}+\sum _{v<_\tau w} F_v*\theta _v,\) 
for some \(\theta _v\in Y)\) . For \(h\in {^{\mathrm {BL}}\mathcal {H}}(T_\) , \(h=\sum _{v\in W^v} T_v*\tilde{\theta }_v\) , write
\(\max \mathrm {supp}(h)=\lbrace w\in W^v| \tilde{\theta }_w\ne 0\text{ and }\forall w^{\prime }\in W^v, w^{\prime }>w, \tilde{\theta }_{w^{\prime }}=0\rbrace .\) 
Then by [1]} and (REF ), \(\max \mathrm {supp}(\tilde{\mathcal {K}}_w)=\lbrace w\rbrace \) , for \(w\in W_{(\tau )}\)  and thus \((\tilde{\mathcal {K}}_{w})_{w\in W_{(\tau )}}\)  is a free family of \({^{\mathrm {BL}}\mathcal {H}}(T_\) .
By (REF ) and (3), \(\sum _{k\in \mathbb {N},s_1,\ldots ,s_k\in {S}_\tau } \mathcal {K}_{s_1}*\ldots *\mathcal {K}_{s_k}*Y)_\tau \subset \bigoplus _{w\in W_{(\tau )}} \tilde{\mathcal {K}}_w*Y)_\tau \) . The converse inclusion is obtained similarly and thus \({K}_\tau =\sum _{k\in \mathbb {N},s_1,\ldots ,s_k\in {S}_\tau } \mathcal {K}_{s_1}*\ldots *\mathcal {K}_{s_k}*Y)_\tau \) .
By [1]}, \({^{\mathrm {BL}}\mathcal {H}}(T__\tau *{K}_\tau \subset {^{\mathrm {BL}}\mathcal {H}}(T__\tau \)  for \(s\in {S}_\tau \) . Therefore \(\tilde{\mathcal {K}}_w\in {^{\mathrm {BL}}\mathcal {H}}(T__\tau \)  for \(w\in W_{(\tau )}\)  and \({K}_\tau \subset {^{\mathrm {BL}}\mathcal {H}}(T__\tau \) . This completes the proof of Proposition REF . \(\square \)
Remark 3.7 In [1]}, we assumed that \(\tau \in \mathcal {U}_ and that \) W=W()\(. The assumption \) W=W()\( is actually useless (without any change in the proofs). The assumption \) U is however used, to ensure that \(\max \mathrm {supp}(\tilde{\mathcal {K}}_{w})=\lbrace w\rbrace \)  (see [1]}), for \(w\in W_{(\tau )}\)  and thus to prove the freeness of \((\tilde{\mathcal {K}}_w)_{w\in W_{(\tau )}}\) .

Action of \({K}_\tau \)  on \(I_\tau (\tau ,\mathrm {gen},W_{(\tau )})\)
We now fix \(\tau \in \mathcal {U}_. Let \) K=wW() Kw *Y)BLH(T\(. In this subsection, we extend the action of \) KBLH on \(I_\tau (\tau ,\mathrm {gen},W_{(\tau )})\)  to an action of \({K}_\tau \)  on \(I_\tau (\tau ,\mathrm {gen},W_{(\tau )})\)  (see Lemma REF ).
Let \(\mathcal {J}_\tau =\lbrace \theta \in Y)_\tau |\tau (\theta )\ne 0\rbrace \).
Lemma 3.8 
Let \(k\in {K}_\tau \) . Then there exists \(\theta \in Y]\setminus \mathcal {J}_\tau \)  such that \(\theta *k\in {^{\mathrm {BL}}\mathcal {H}}_.\)
Let \(w\in W_{(\tau )}\) . We assume that for all \(v\in W_{(\tau )}\)  such that \(v<_\tau w\) , for all \(\tilde{\theta }\in Y)_\tau \) , there exists \(\theta \in Y]\setminus \mathcal {J}_\tau \)  such that \(\theta *\tilde{\mathcal {K}}_v*\tilde{\theta }\in {^{\mathrm {BL}}\mathcal {H}}_. Let \) Y)\(. Let \) sS\( be such that \) v:=sw<w\(. Let \) 1Y]J\( be such that \) 1*Kv*BLH. Let \(\theta _2\in Y]\setminus \mathcal {J}_\tau \)  be such that \(\tilde{\mathcal {K}}_s*\theta \in {^{\mathrm {BL}}\mathcal {H}}_, which exists by Proposition~\ref {propBL_relations}(\ref {itAlgebra_K}). Set \) 3=(1*2)*s( 1*2)Y]J\(. Then \) s3=3\(, thus \) s(3)=0\( and hence \) 3*Ks=Ks*3\(. Therefore \)\(\theta _3* \tilde{\mathcal {K}}_{w}*\theta =\tilde{\mathcal {K}}_s*\theta _3*\tilde{\mathcal {K}}_v*\theta =\tilde{\mathcal {K}}_s *\theta _2 *{^s (\theta _1*\theta _2)}*\theta _1*\tilde{\mathcal {K}}_v\in {^{\mathrm {BL}}\mathcal {H}}_\)\( We deduce that for all \) wW()\( and \) Y)\(, there exists \) Y]J\( such that \) *Kw*BLH.
Let now \(k\in {K}_\tau \) . Write \(k=\sum _{w\in W_{(\tau )}} \tilde{\mathcal {K}}_w *\tilde{\theta }_w\) , with \((\tilde{\theta }_w)\in (Y)_\tau )^{(W_{(\tau )})}\) . For \(w\in W_{(\tau )}\) , choose \(\theta _w\in Y]\setminus \mathcal {J}_\tau \)  such that \(\theta _w*\tilde{\mathcal {K}}_w*\tilde{\theta }_w\in {^{\mathrm {BL}}\mathcal {H}}_. Set \) =wW()|w0 wY]J\(. Then \) *kBLH, which proves the lemma.
Lemma 3.9 
Let \(\theta \in Y]\setminus \mathcal {J}_\tau \) . Then the map \(m_\theta :I_\tau (\tau ,\mathrm {gen},W_{(\tau )})\rightarrow I_\tau (\tau ,\mathrm {gen},W_{(\tau )})\)  defined by \(m_\theta (x)=\theta .x\) , for \(x\in I_\tau (\tau ,\mathrm {gen},W_{(\tau )})\) , is injective.

Let \(x\in I_\tau \) . Write \(x=\sum _{w\in W^v} a_w T_w.\mathbf {v}_\tau \) , where \((a_w)\in {(W^v)}\) . Let \(w\in W^v\)  be such that \(a_w\ne 0\)  and such that for all \(v\in W^v\)  such that \(a_v\ne 0\) , we have \(v\lnot \ge w\) . Then by [1]}, we have \(\theta .x-a_w T_w*{^{w^{-1}}\theta }.\mathbf {v}_\tau \in \bigoplus _{v\in W^v, v\lnot \ge w}T̏_v.\mathbf {v}_\tau \) . By [1]}, \(w\in W_{(\tau )}\subset W_\tau \) . Therefore \({^{w^{-1}}\theta }.\mathbf {v}_\tau =\tau (\theta ).\mathbf {v}_\tau \ne 0\) , which proves that \(\theta .x\ne 0\) . Therefore \(m_\theta \)  is injective.
Lemma 3.10 
We have the following properties:

\(I_\tau (\tau ,\mathrm {gen},W_{(\tau )})=({K}_\tau \cap {^{\mathrm {BL}}\mathcal {H}}_.\mathbf {v}_\tau \) ,

\({K}_\tau \cap {^{\mathrm {BL}}\mathcal {H}}_ stabilizes \) I(,gen,W())\( and the action of \) KBLH on \(I_\tau (\tau ,\mathrm {gen},W_{(\tau )})\)  extends uniquely to an action of \({K}_\tau \)  on \(I_\tau (\tau ,\mathrm {gen},W_{(\tau )})\) . This action is as follows. Let \(x\in I_\tau (\tau ,\mathrm {gen},W_{(\tau )})\)  and \(k\in {K}_\tau \) . Write \(x=h.x\) , with \(h\in {^{\mathrm {BL}}\mathcal {H}}_{K}_\tau \) . Write \(k*h=\sum _{w\in W_{(\tau )}} \tilde{\mathcal {K}}_w \tilde{\theta }_w\) , with \((\theta _w)\in (Y)_\tau )^{(W_{(\tau )})}\) . Then:
\(k.x=k*h.\mathbf {v}_\tau =\sum _{w\in W_{(\tau )}} \tau (\tilde{\theta })\tilde{\mathcal {K}}_w (\tau ).\mathbf {v}_\tau .\) 


By [1]}, we have \(I_\tau (\tau ,\mathrm {gen},W_{(\tau )})=\mathrm {ev}_\tau ({K}_\tau ).\mathbf {v}_\tau \) .
For \(w\in W_{(\tau )}\) , write \(\tilde{\mathcal {K}}_{w}=\sum _{v\in W^v} T_w *\theta _{v,w}\) , with \(\theta _{v,w}\in Y)\)  for \(v\in W^v\)  and \(\lbrace v\in W^v|\theta _{v,w}\ne 0\rbrace \)  finite. By [1]}, we can write \(\theta _{v,w}=\frac{f_{v,w}}{g_{v,w}}\) , where \(f_{v,w},g_{v,w}\in Y]\)  and \(\tau (g_{v,w})\ne 0\)  for \(v\in W^v\)  such that \(f_{v,w}\ne 0\) . For \(w\in W_{(\tau )}\) , set \(g_w=\prod _{v\in W^v|f_{v,w}\ne 0} g_{v,w}\in Y]\) . Then
\(\mathrm {ev}_\tau (\tilde{\mathcal {K}}_w)=\mathrm {ev}_\tau (\frac{1}{\tau (g_w)}\tilde{\mathcal {K}}_w g_w)\text{ and }\frac{1}{\tau (g_w)}\tilde{\mathcal {K}}_w g_w\in {K}_\tau \cap {^{\mathrm {BL}}\mathcal {H}}_\) 
Therefore
\(I_\tau (\tau ,\mathrm {gen},W_{(\tau )})=({K}_\tau \cap {^{\mathrm {BL}}\mathcal {H}}_.\mathbf {v}_\tau ,\) 
which proves (1).
(2) By Proposition REF (REF ), \({K}_\tau \cap {^{\mathrm {BL}}\mathcal {H}}_ is a subalgebra of \) BLH, which proves that \({K}_\tau \cap {^{\mathrm {BL}}\mathcal {H}}_ stabilizes \) I(,gen,W())\(. For \) wW()\(, set \) Lw=Kw*gwBLHK\( and set \)\({L}_\tau =\bigoplus _{w\in W_{(\tau )}} L_w*Y]\subset {^{\mathrm {BL}}\mathcal {H}}_{K}_\tau .\)\(\)
Let \(\tilde{\tau }:Y)_\tau \rightarrow  be the algebra morphism extending \) :Y]. Let \(\tilde{M}=\mathrm {Ind}_{Y)_\tau }^{{K}_\tau }(\tilde{\tau })\)  be the representation of \({K}_\tau \)  obtained by inducing \(\tilde{\tau }\)  to \({K}_\tau \) . Then
\(\tilde{M}=\bigoplus _{w\in W_{(\tau )}} \tilde{\mathcal {K}}_w.\textbf {v}_{\tilde{\tau }},\) 
where \(\textbf {v}_{\tilde{\tau }}\in \tilde{M}\)  is such that \(\theta .\textbf {v}_{\tilde{\tau }}=\tilde{\tau }(\theta ).\textbf {v}_{\tilde{\tau }}\) , for \(\theta \in Y)_\tau \) . Let \(\psi :\tilde{M}\rightarrow I_\tau (\tau ,\mathrm {gen},W_{(\tau )})\)  be defined by \(\psi (k.\textbf {v}_{\tilde{\tau }})=k.\mathbf {v}_\tau \) , for \(k\in {L}_\tau \) . Let us prove that \(\psi \)  is well defined. Let \(k\in {L}_\tau \)  be such that \(k.\textbf {v}_{\tilde{\tau }}=0\) . We can write \(k=\sum _{w\in W^v} L_w*\theta _w\) , for some \((\theta _w)\in Y]^{(W_{(\tau )})}\) . Then \( \tau (\theta _w)=0\)  for all \(w\in W_{(\tau )}\)  and thus \(k.\mathbf {v}_\tau =0\) , which proves that \(\psi :{L}_\tau .\textbf {v}_{\tilde{\tau }}\rightarrow I_\tau (\tau ,\mathrm {gen},W_{(\tau )})\)  is well defined. Moreover, \(\tilde{M}=\bigoplus _{w\in W_{(\tau )}}_w.\textbf {v}_{\tilde{\tau }}=\bigoplus _{w\in W_{(\tau )}} L̏_w.\textbf {v}_{\tilde{\tau }}={L}_\tau .\textbf {v}_{\tilde{\tau }}\) . Thus \(\psi :\tilde{M}\rightarrow I_\tau (\tau ,\mathrm {gen},W_{(\tau )})\)  is well defined. Then \(\psi \)  is an isomorphism of \({L}_\tau \) -modules. We equip \(I_\tau (\tau ,\mathrm {gen},W_{(\tau )})\)  with the structure of a \({K}_\tau \) -module by setting \(k\odot x=\psi \big (k.\psi ^{-1}(x)\big )\)  for \(k\in {K}_\tau \)  and \(x\in I_\tau (\tau ,\mathrm {gen},W_{(\tau )})\) .
Let \(w\in W_{(\tau )}\)  and \(\theta \in Y)_\tau \) . Then
\((\tilde{\mathcal {K}}_w*\theta )\odot \mathbf {v}_\tau =\psi (\tilde{\mathcal {K}}_w*\theta .\textbf {v}_{\tilde{\tau }})=\tau (\theta )\psi (\tilde{\mathcal {K}}_w.\textbf {v}_{\tilde{\tau }})=\tau (\theta )\tilde{\mathcal {K}}_w\odot \mathbf {v}_\tau .\) 
By applying this to \(\theta =g_w\) , we deduce that
\(\tilde{\mathcal {K}}_w\odot \mathbf {v}_\tau =\frac{1}{\tau (g_w)}\tilde{\mathcal {K}}_w*g_w\odot \mathbf {v}_\tau =\frac{1}{\tau (g_w)}\tilde{\mathcal {K}}_w*g_w.\mathbf {v}_\tau =\frac{1}{\tau (g_w)}\mathrm {ev}_\tau (\tilde{\mathcal {K}}_w*g_w).\mathbf {v}_\tau =\tilde{\mathcal {K}}_w(\tau ).\mathbf {v}_\tau .\) 
Combining (REF ) and (REF ) yields (REF ).
Let \(k\in {^{\mathrm {BL}}\mathcal {H}}_{K}_\tau \) . Write \(k=\sum _{w\in W_{(\tau )}} \tilde{\mathcal {K}}_w*\theta _w\) , with \((\theta _w)\in (Y)_\tau )^{(W_{(\tau )})}\) . For \(w\in W_{(\tau )}\) , write \(\tilde{\mathcal {K}}_w=\sum _{v\in W^v}T_v*\theta _{v,w}\) , with \((\theta _{v,w})_{v\in W^v}\in (Y)_\tau )^{(W^v)}\) . Then \(k=\sum _{v\in W^v}\sum _{w\in W_{(\tau )}} T_v*\theta _{v,w}*\theta _w\)  and \(\sum _{w\in W_{(\tau )}} \theta _{v,w}*\theta _{w}\in Y]\)  for every \(v\in W^v\) . Then
\(\begin{aligned} k.\mathbf {v}_\tau &=\sum _{v\in W^v} \tau (\sum _{w\in W_{(\tau )}} \theta _{v,w}*\theta _w) T_v.\mathbf {v}_\tau \\&=\sum _{v\in W^v} \sum _{w\in W_{(\tau )}}\tau (\theta _{v,w}*\theta _w) T_v.\mathbf {v}_\tau \\&= \sum _{w\in W_{(\tau )}} \tau (\theta _w)\mathrm {ev}_\tau (\sum _{v\in W^v} T_v*\theta _{v,w}).\mathbf {v}_\tau \\&= \sum _{w\in W_{(\tau )}} \tau (\theta _w) \mathrm {ev}_\tau (\tilde{\mathcal {K}}_w).\mathbf {v}_\tau =k\odot \mathbf {v}_\tau ,\end{aligned}\) 
by (REF ). Therefore \(\odot \)  extends the action \(.\)  of \({^{\mathrm {BL}}\mathcal {H}}_{K}_\tau \)  on \(I_\tau (\tau ,\mathrm {gen},W_{(\tau )})\)  to an action of \({K}_\tau \)  on \(I_\tau (\tau ,\mathrm {gen},W_{(\tau )})\) .
We now prove the uniqueness of such an action. Let \(\boxdot \)  be an action of \({K}_\tau \)  on \(I_\tau (\tau ,\mathrm {gen},W_{(\tau )})\)  such that \(k\boxdot x=k.x\)  for all \(k\in {^{\mathrm {BL}}\mathcal {H}}_{K}_\tau \)  and \(x\in I_\tau (\tau ,\mathrm {gen},W_{(\tau )})\) . Let \(k\in {K}_\tau \) . Let \(\theta \in Y]\setminus \mathcal {J}_\tau \)  be such that \(\theta *k\in {^{\mathrm {BL}}\mathcal {H}}_, which exists by Lemma~\ref {lemLeft_multiplication_CY_AC}. Then \)\((\theta *k).x=(\theta *k)\boxdot x=\theta \boxdot (k\boxdot \mathbf {v}_\tau )=\theta .(k\boxdot \mathbf {v}_\tau ).\)\( By Lemma~\ref {lemInjectivity_multiplication_theta} we deduce that \) kx=m-1(*k.x)\(. This proves that \) =\( and concludes the proof of this lemma.\)
Examples of \(\tau \)  for which \({S}_\tau \)  is infinite
The set \(\Phi ^\vee \)  is a root system in the sense of [30]}. Then \(\Phi _{(\tau )}^\vee \)  is a subroot system of \(\Phi ^\vee \)  in the sense of [30]}. It is proved in [30]} that the bases of a subroot system of a root system admitting a finite basis need not be finite. However, we do not know if the root system given in this example is of the form \(\Phi ^\vee _{(\tau )}\) , for some \(\tau \in T_. Suppose that \) s=s'\( for all \) sS\(, then \) ()={|()=1}\(. Thus if \) T, \(\Phi ^\vee _{(\tau )}\)  is closed in the sense that for all \(\alpha ^\vee ,\beta ^\vee \in \Phi ^\vee _{(\tau )}\) , if \(\alpha ^\vee +\beta ^\vee \in \Phi ^\vee \) , then \(\alpha ^\vee +\beta ^\vee \in \Phi _{(\tau )}^\vee \) . We may ask whether this property ensures the finiteness of \({S}_\tau \) . We prove below that this is not the case.
For \(w\in W^v\) , set \(N_{\Phi ^\vee }(w)=\lbrace \alpha ^\vee \in \Phi ^\vee _+|w.\alpha ^\vee \in \Phi ^\vee _-\rbrace \).
Lemma 3.11 
Let \(A=(a_{i,j})_{i,j\in \llbracket 1,4\rrbracket }\)  be an invertible Kac-Moody matrix such that for all \(i,j\in \llbracket 1,4\rrbracket \) , \(a_{i,j}\le -2\) , for all \((i,j)\in \llbracket 1,4\rrbracket \setminus \lbrace (4,3)\rbrace \) , \(a_{i,j}\)  is even and \(a_{4,3}\)  is odd. We assume that \(\mathbb {A}=\bigoplus _{i=1}^4 \mathbb {R}\alpha _i^\vee \) , which is possible since \(A\)  is invertible by [33]}. Let \(\mathrm {ht}:\mathbb {A}\rightarrow \mathbb {R}\)  be defined by \(\mathrm {ht}(\sum _{i=1}^4 n_i\alpha _i^\vee )=\sum _{i=1}^4 n_i\) , for \((n_i)\in \mathbb {R}^4\) . Let \(q\)  be a prime power and set \(\sigma _i=\sigma _i^{\prime }=\sqrt{q}\)  for \(i\in \llbracket 1,4\rrbracket \) . Let \(\tau : Y\rightarrow \lbrace -1,1\rbrace \)  be defined by \(\tau (\lambda )=(-1)^{\mathrm {ht}(\lambda )}\)  for \(\lambda \in Y\) . Let \(W^{\prime }=\langle r_1,r_2\rangle \subset W^v\) . Then \(\lbrace w r_3r_4r_3 w^{-1}|w\in W^{\prime }\rbrace \subset {S}_\tau \) . In particular, \({S}_\tau \)  is infinite.

Let \(w\in W^{\prime }\)  and \(v=wr_3r_4r_3w^{-1}\) . Then by [34]}, \(wr_3.\alpha _4^\vee \in N_{\Phi ^\vee }(v)\)  and \(N_{\Phi ^\vee }(v)\setminus \lbrace wr_3.\alpha _4^\vee \rbrace \subset \lbrace u.\alpha _i^\vee |u\in W^v, i\in \llbracket 1,3\rrbracket \rbrace \) . We have \(r_3.\alpha _4^\vee =\alpha _4^\vee -a_{4,3}\alpha _3^\vee \) . Moreover if \(i\in \llbracket 1,2\rrbracket \) , \(r_i(\alpha _4^\vee -a_{4,3}\alpha _3^\vee +2Q^\vee )\subset \alpha _4^\vee -a_{4,3}\alpha _3^\vee +2Q^\vee \) . In particular, \(wr_3.\alpha _4^\vee \in \alpha _4^\vee -a_{4,3}\alpha _3^\vee +2Q^\vee \)  and \(\tau (wr_3.\alpha _4^\vee )=1\) . Now if \(i\in \llbracket 1,3\rrbracket \)  and \(j\in \llbracket 1,4\rrbracket \) , \(r_j.(\alpha _i^\vee +2Q^\vee )\subset \alpha _i^\vee +2Q^\vee \) . Therefore \(\tau (u.\alpha _i^\vee )=-1\)  if \(u\in W^v\) . In particular, \(N_{\Phi ^\vee }(v)\cap \Phi _{(\tau )}=\lbrace wr_3.\alpha _4^\vee \rbrace =\lbrace \alpha _v^\vee \rbrace \)  and by [1]}, \(v\in {S}_\tau \) . Using [34]} we deduce that \({S}_\tau \)  is infinite.

Kato's irreducibility criterion
Let \(\tau \in \mathcal {U}_. Let \) xI(,gen,W())\(. Write \) x=wW() aw Kw.v\(, with \) (aw)(Wv)\(. Let \) supp(x)={wW()|aw0}\( and \) (x)={(w)|wS}\(. We set \) (0)=-\(.\)
Let \(x\in I_\tau (\tau ,\mathrm {gen},W_{(\tau )})\) . We define \(\mathrm {ord}_\tau (x)\)  as the minimum of the \(k\in \mathbb {N}\)  such that for all \(\theta _1,\ldots ,\theta _k\in \mathcal {J}_\tau \) , we have \(\theta _1\ldots \theta _k.x=0\) . We will see in Lemma REF  that \(\mathrm {ord}_\tau (x)\in \mathbb {N}\)  for \(x\in I_\tau (\tau ,\mathrm {gen},W_{(\tau )})\) . The aim of this subsection is to prove that \(I_\tau (\tau ,W_{(\tau )}):=I_\tau (\tau ,\mathrm {gen},W_{(\tau )})\cap I_\tau (\tau )=_\tau \)  (see Theorem REF ). We actually prove that for all \(x\in I_\tau (\tau ,\mathrm {gen},W_{(\tau )})\) , \(\mathrm {ord}_\tau (x)=\ell _\tau (x)+1\) . We then deduce Kato's irreducibility criterion (see Corollary REF ).
For \(s\in {S}_\tau \) , we set
\(\sigma _{s,\tau }^{\prime \prime }=\frac{1}{2}\big ((\sigma _s^2-1)+\sigma _s(\sigma ^{\prime }_s-\sigma _s^{\prime -1})\tau (\alpha _s^\vee )\big ).\) 
 When \(\sigma _s=\sigma _s^{\prime }\) , then \(\tau (\alpha _s^\vee )=1\)  and \(\sigma _{s,\tau }^{\prime \prime }=\sigma _s^2-1\) .
As we assumed that \(|\sigma _s|,|\sigma _s^{\prime }|>1\) , for all \(s\in {S}\) , we have \(\sigma _{s,\tau }^{\prime \prime }\ne 0\)  for every \(s\in {S}_\tau \) . Indeed, we have \(\sigma _{s,\tau }^{\prime \prime }=0\)  if and only if \(\sigma _s=\sigma ^{\prime }_s=\pm 1\)  or \(\sigma ^{\prime }_s\ne \sigma _s\)  and \(\sigma _s^{\prime }=-\sigma _s^{-1}\) . We have \({S}_\tau \subset {R}_{(\tau )}\)  and thus \(\tau (\alpha _s^\vee )\in \lbrace -1,1\rbrace \) . If \(\tau (\alpha _s^\vee )=-1\) , then \(\sigma _s\ne \sigma _s^{\prime }\)  and \(\sigma _s^2-1+\sigma _s(\sigma _s^{\prime }-\sigma _s^{\prime -1})\tau (-\alpha _s^\vee )=\sigma _s^2-1-\sigma _s(\sigma _s^{\prime }-\sigma _s^{\prime -1})=0\)  if and only if \(\sigma _s^{\prime }\in \lbrace \sigma _s,-\sigma _s^{-1}\rbrace \)  if and only if \(\sigma _s^{\prime }=-\sigma _s^{-1}\) .
Lemma 4.1 
Let \(s\in {S}_\tau \)  and \(\lambda \in Y\) . Then
\(\tau \big (\tilde{\Omega }_s(Z^\lambda )\big )=\tau (\lambda )\sigma _{s,\tau }^{\prime \prime }\alpha _s(\lambda ).\) 

We have
\(\tilde{\Omega }_s(Z^\lambda )=\big ((\sigma _s^2-1)+\sigma _s(\sigma _s^{\prime }-\sigma _s^{\prime -1})Z^{-\alpha _s^\vee }\big ) \frac{1-Z^{-\alpha _s(\lambda )\alpha _s^\vee }}{1-Z^{-2\alpha _s^\vee }}Z^{\lambda }.\) 
We have \(\tau (\alpha _s^\vee )\in \lbrace -1,1\rbrace \) . If \(\alpha _s(\lambda )\)  is odd, then \(\sigma _s=\sigma _s^{\prime }\)  and \(\tau (\alpha _s^\vee )=1\)  (since the denominator of \(Q_s^T\)  is then \(1-Z^{-\alpha _s^\vee }\) ). We then conclude with Remark REF  (REF ). If \(\alpha _s(\lambda )\)  is even, a computation similar to the one of Remark REF  (REF ) enables to prove (REF ).
For \(h\in {K}_\tau \) , \(h=\sum _{w\in W_{(\tau )}} \tilde{\mathcal {K}}_w*\theta _w\) , with \((\theta _w)\in {(W_{(\tau )})}\)  we set \(\ell _\tau (h)=\max \lbrace \ell _\tau (w)|\theta _w\ne 0\rbrace \) .
Lemma 4.2 
Let \(w\in W_{(\tau )}\) . Fix a reduced writing \(w=s_1\ldots s_k\)  of \(w\) , with \(k=\ell _\tau (w)\)  and \(s_1,\ldots ,s_k\in {S}_\tau \) . For \(i\in \llbracket 1,k\rrbracket \) , set \(v_i=s_1\ldots s_{i-1}\) , \(\tilde{v}_i=s_{i+1}\ldots s_k\) . Let \(E=\lbrace i\in \llbracket 1,k\rrbracket |\ell _\tau (v_i\tilde{v}_i)=k-1\rbrace \) . Let \(\theta \in Y)\) . Then there exists \(h\in {K}_\tau \)  such that \(\ell _\tau (h)\le k-2\)  and
\(\theta *\tilde{\mathcal {K}}_w=\tilde{\mathcal {K}}_w*{{^{w^{-1}}\theta }}+\sum _{i\in E}\tilde{\mathcal {K}}_{v_i\tilde{v_i}} *{^{\tilde{v}_i^{-1}}\tilde{\Omega }_{s_i}({^{v_i^{-1}}\theta }}) +h.\) 
In particular if \(\theta \in Y)_\tau \)  we have
\(\theta .\tilde{\mathcal {K}}_w.\mathbf {v}_\tau =\tau (\theta ) \tilde{\mathcal {K}}_w.\mathbf {v}_\tau +\tau (\lambda )\sum _{i\in E} \sigma _{s_i,\tau }^{\prime \prime }\alpha _{s_i}(v_i^{-1}.\theta ) \tilde{\mathcal {K}}_{v_i\tilde{v}_i}.\mathbf {v}_\tau +h.\mathbf {v}_\tau .\) 

We prove it by induction on \(k\) . If \(k=0\) , this is clear. We assume \(k\ge 1\) . Set \(w^{\prime }=ws_k\)  and \(\tilde{v}_i^{\prime }=v_is_k\)  for \(i\in \llbracket 1,k-1\rrbracket \) . Assume that we can write
\(\theta *\tilde{\mathcal {K}}_{w^{\prime }}=\tilde{\mathcal {K}}_{w^{\prime }}*{^{w^{\prime -1}}\theta }+\sum _{i=1}^{k-1} \tilde{\mathcal {K}}_{v_i\tilde{v^{\prime }_i}} *{^{\tilde{v^{\prime }}_i^{-1}}\tilde{\Omega }_{s_i}(^{v_i^{-1}}\theta }) +h^{\prime },\) 
where \(h^{\prime }\in {^{\mathrm {BL}}\mathcal {H}}(T_\)  and \(\ell _\tau (h^{\prime })\le k-3\) .
By Proposition REF  (REF ), we have
\(\begin{aligned} \theta *\tilde{\mathcal {K}}_w &= \big (\tilde{\mathcal {K}}_{w^{\prime }}*{^{w^{\prime -1}}\theta }+\sum _{i=1}^{k-1} \tilde{\mathcal {K}}_{v_i\tilde{v^{\prime }_i}} *{^{\tilde{v^{\prime }}_i^{-1}}\tilde{\Omega }_{s_i}(^{v_i^{-1}}\theta }) \big )*\tilde{\mathcal {K}}_{s_k}\\ &=\tilde{\mathcal {K}}_{w}*{^{w^{-1}}\theta } +\tilde{\mathcal {K}}_{w^{\prime }}*\tilde{\Omega }_{s_k}({^{w^{\prime -1}}\theta }) +\sum _{i=1}^{k-1}\left( \tilde{\mathcal {K}}_{v_i\tilde{v_i}} *{^{\tilde{v}_i^{-1}}\tilde{\Omega }_{s_i}({^{v_i^{-1}}\theta }})\right)+h^{\prime }*\tilde{\mathcal {K}}_{s_k}+h^{\prime \prime }\\&=\tilde{\mathcal {K}}_{w}*{^{w^{-1}}\theta } +\sum _{i=1}^{k} \tilde{\mathcal {K}}_{v_i\tilde{v_i}} *{^{\tilde{v}_i^{-1}}\tilde{\Omega }_{s_i}({^{v_i^{-1}}\theta }})+ +h^{\prime }*\tilde{\mathcal {K}}_{s_k}+h^{\prime \prime },\end{aligned}\) 
where
\(h^{\prime \prime } =\sum _{i=1}^{k-1}\tilde{\mathcal {K}}_{v_i\tilde{v_i^{\prime }}} *\tilde{\Omega }_{s_k}\left({^{\tilde{v}_i^{-1}}\tilde{\Omega }_{s_i}({^{v_i^{-1}}\theta }})\right)\in \bigoplus _{v\in W_{(\tau )}|\ell _\tau (v)\le k-2} \tilde{\mathcal {K}}_{v}*Y)_\tau . \) 
Moreover by Proposition REF , \(\ell _\tau (h^{\prime }*\tilde{\mathcal {K}}_{s_k})\le k-1\) . By eliminating the \(i\)  such that \(\ell _\tau (v_i\tilde{v}_i)<k-1\)  in (REF ), we deduce (REF ).
Using Lemma REF , we deduce (REF ).
Lemma 4.3 
Let \(x\in I_\tau (\tau ,\mathrm {gen},W_{(\tau )})\) , \(\theta \in \mathcal {J}_\tau \)  and \(w\in W^v\) . Then \(\ell _\tau (\theta .x)\le \ell _\tau (x)-1\) . In particular we have \(\mathrm {ord}_\tau (x)\le \ell _\tau (x)+1\)  for all \(x\in I_\tau (\tau ,\mathrm {gen},W_{(\tau )})\) .

This follows from (REF ).
Lemma 4.4 
We assume that there exists \(\rho \in *\)  such that
\(\sigma ^{\prime \prime }_{s,\tau }\in \rho \mathbb {R}^*_+,\) 
for all \(s\in {S}\) . Let \(w\in W_{(\tau )}\) . Then \(\mathrm {ord}_\tau (\tilde{\mathcal {K}}_w.\mathbf {v}_\tau )=\ell _\tau (w)+1\) .

Pick \(\lambda \in C^v_f\)  and set \(\theta =Z^\lambda -\tau (\lambda )\in \mathcal {J}_\tau \) . For \(v\in W_{(\tau )}\) , we define \(C_v\in  by \)\(\theta ^{\ell _\tau (v)}.\mathbf {v}=C_v.\mathbf {v}_\tau ,\)\( which is well defined by Lemma~\ref {lemLength_decreasing_multiplication_theta}. Fix a reduced writing \) v=s1...sk\( of \) w\(, with \) k=(w)\( and \) s1,...,skS\(. We want to prove that \) Cw0\(. We assume that for all \) vW()\( such that \) (v)<(w)\(, we have \) Cv(())(v)R*+\(. For \) i1,k\(, set \) vi=s1...si-1\(, \) vi=si+1...sk\(. Let \) E={i1,k|(vivi)=k-1}\(.Then by Lemma~\ref {lemComputation_terms ordern-1_commutation} and (\ref {eqFormula_tau_Omega_s}), we have \)\(\begin{aligned}C_w &=\sum _{i\in E} \tau \big (\tilde{\Omega }_{s_i}(^{v_i^{-1}}\theta )\big )C_{v_i\tilde{v}_i}\\&= \tau (\lambda )\sum _{i\in E}\alpha _{s_i}(\lambda ) \sigma _{s_i,\tau }^{\prime \prime } v_i.\alpha _{s_i}(\lambda ) C_{v_i\tilde{v}_i}. \end{aligned} \)\( If \) iE\(, we have \) (visi)>(vi)\( and thus by \cite [1.3.13 Lemma]{kumar2002kac}, \) vi.si+\(. Therefore \) vi.si()Z>0\(. We deduce that \) Cw(())kR*+\(. In particular, \) Cw0\( and \) ord(Kw .v)(w)+1\(. By Lemma~\ref {lemLength_decreasing_multiplication_theta} we deduce that \) ord(Kw .v)=(w)+1\(.\)
Lemma 4.5 
Let \(x\in I_\tau (\tau ,\mathrm {gen},W_{(\tau )})\)  and \(s\in {S}_\tau \) . Then \(\mathrm {ord}_\tau (\tilde{\mathcal {K}}_s.x)\le \mathrm {ord}_\tau (x)+1\) .

We prove it by induction on \(\mathrm {ord}_\tau (x)\) . If \(\mathrm {ord}_\tau (x)=0\) , then \(x=0\)  and it is clear. We assume that \(\mathrm {ord}_\tau (x)>0\)  and that for all \(y\in I_\tau (\tau ,\mathrm {gen},W_{(\tau )})\)  such that \(\mathrm {ord}_\tau (y)<\mathrm {ord}_\tau (x)\) , we have \(\mathrm {ord}_\tau (\tilde{\mathcal {K}}_s.y)\le \mathrm {ord}_\tau (y)+1\) . Let \(\theta \in \mathcal {J}_\tau \) . By Proposition REF  (REF ), we have
\(\theta .\tilde{\mathcal {K}}_s.x=\tilde{\mathcal {K}}_s.{^s\theta }.x+\tilde{\Omega }_s(\theta ).x,\) 
for \(\theta \in \mathcal {J}_\tau \) . Moreover \(\mathrm {ord}_\tau ({^s\theta }.x)< \mathrm {ord}_\tau (x)\)  and thus \(\mathrm {ord}_\tau (\tilde{\mathcal {K}}_s.{^s\theta }.x)\le \mathrm {ord}_\tau ({^s\theta }.x)+1\le _\tau \mathrm {ord}_\tau (x)\)  by assumption. Moreover, \(\mathrm {ord}_\tau \left(\tilde{\Omega }_s(\theta ).x\right)\le \mathrm {ord}_\tau (x)\) , therefore \(\mathrm {ord}_\tau (\theta .\tilde{\mathcal {K}}_s.x)\le \mathrm {ord}_\tau (x)\)  for every \(\theta \in \mathcal {J}_\tau \)  and thus \(\mathrm {ord}_\tau (\tilde{\mathcal {K}}_s.x)\le \mathrm {ord}_\tau (x)+1\) . Lemma follows.
Let \(x\in I_\tau (\tau ,\mathrm {gen},W_{(\tau )})\setminus \lbrace 0\rbrace \) . Write \(x=\sum _{w\in W_{(\tau )}}a_w\tilde{\mathcal {K}}_w.\mathbf {v}_\tau \) . Let \(M=\lbrace w\in \mathrm {supp}(x)|\ell _\tau (w)=\ell _\tau (x)\rbrace \) , \(\mathcal {N}_\tau =|M|\)  and \(\mathrm {LT}(x)=\sum _{w\in M} a_w \tilde{\mathcal {K}}_w.\mathbf {v}_\tau \) .
Theorem 4.6 
Let \(\tau \in \mathcal {U}_. We assume that (\ref {eqDef_rho}) is satisfied and that \) |s|,|s'|>1\(, for all \) sS\(. Let \) xI(,gen,W()){0}\(. Then \) ord(x)=(x)+1\(. In particular, \) I(,gen,W())I()=\( and \cite [Conjecture 5.16]{hebert2021decompositions} is true.\)
If \(\mathcal {N}_\tau (x)=1\) , this is Lemma REF  and if \(\ell _\tau (x)=0\) , this is clear. We now assume that \(\mathcal {N}_\tau (x)\ge 2\)  and that for all \(y\in I_\tau (\tau ,\mathrm {gen},W_{(\tau )})\setminus \lbrace 0\rbrace \)  such that \(\ell _\tau (y)<\ell _\tau (x)\)  or \(\mathcal {N}_\tau (y)<\mathcal {N}_\tau (x)\) , we have \(\mathrm {ord}_\tau (y)=\ell _\tau (y)+1\) . Let \(M=\mathrm {supp}\big (\mathrm {LT}(x)\big )\) . There are two cases:

there exists \(s\in {S}\)  such that for all \(w\in M\) , \(sw<_{\tau } w\) ,

for all \(s\in {S}\) , there exists \(w\in M\)  such that \(sw>_\tau w\) .

We first assume (1). Then we can write
\(x=\tilde{\mathcal {K}}_s.x^{\prime }+x^{\prime \prime },\) 
with \(\ell _\tau (x^{\prime })=\ell _\tau (x)-1\) , \(\ell _\tau (x^{\prime \prime })\le \ell _\tau (x^{\prime })\)  and \(sv>_\tau v\)  for all \(v\in \mathrm {supp}(x^{\prime })\) . Let \(\theta \in \mathcal {J}_\tau \) . Then
\( \theta .\tilde{\mathcal {K}}_s.x=\tilde{\mathcal {K}}_s*{^{s}\theta }.x^{\prime }+\tilde{\Omega }_s(\theta ).x^{\prime }+\theta .x^{\prime \prime }.\) 
By Lemma REF , we have \(\ell _\tau ({^s\theta }.x^{\prime }) \le \ell _\tau (x^{\prime })-1\) . For all \(v\in \mathrm {supp}({^s\theta }.x^{\prime })\) , if \(sv<_\tau v\) , then \(\ell _\tau (\tilde{\mathcal {K}}_s*\tilde{\mathcal {K}}_v.\mathbf {v}_\tau )\le \ell _\tau (v)\le \ell _\tau ({^s\theta }.x^{\prime })\le \ell _\tau (x^{\prime })-1\)  by Proposition REF  (). Therefore for all \(v\in \mathrm {supp}({\tilde{\mathcal {K}}_s*^s\theta }.x)\)  such that \(\ell _\tau (v)=\ell _\tau (x^{\prime })\) , we have \(sv<_\tau v\) . Therefore
\(\mathrm {supp}\big (\mathrm {LT}(\tilde{\mathcal {K}}_s*{^s\theta }*x^{\prime })\big )\cap \mathrm {supp}\big (\mathrm {LT}(x^{\prime })\big )=\emptyset .\) 
Take \(\lambda \in Y\)  such that \(\alpha _s(\lambda )=1\)  and set \(\theta =Z^\lambda -\tau (\lambda )\in \mathcal {J}_\tau \) . Then by Lemma REF , \(\tau \big (\tilde{\Omega _s}(\theta )\big )=\tau (\lambda )\sigma _{s,\tau }^{\prime \prime }\alpha _s(\lambda )=\tau (\lambda )\sigma _{s,\tau }^{\prime \prime }\)  and thus by combining (REF ) and (REF ), we obtain that \(\ell _\tau (\theta .\tilde{\mathcal {K}}_s.x)=\ell _\tau (x^{\prime })\) . By assumption we deduce that \(\mathrm {ord}_\tau (\theta .\tilde{\mathcal {K}}_s.x)=\ell _\tau (x^{\prime })+1\) . Consequently \(\mathrm {ord}_\tau (\tilde{\mathcal {K}}_s.x)\ge \ell _\tau (x)+1\)  and by Lemma REF  we deduce that \(\mathrm {ord}_\tau (\tilde{\mathcal {K}}_s.x)=\ell _\tau (x)+1\) .
We now assume that we are in the case (2). Let \(w\in \mathrm {supp}\big ((\mathrm {LT}(x)\big )\)  and \(s\in {S}\)  be such that \(sw<_\tau w\) . Set \(x^{\prime }= \tilde{\mathcal {K}}_s.x\) . Then by assumption we have \(\ell _\tau (x^{\prime })=\ell _\tau (x)+1\)  and thus
\(\mathrm {supp}\big (\mathrm {LT}(x^{\prime })\big )\subset s.\mathrm {supp}\left(\mathrm {LT}(x)\right)\setminus \lbrace sw\rbrace .\) 
In particular \(\mathcal {N}_\tau (x^{\prime })<\mathcal {N}_\tau (x)\) . By our induction assumption we deduce that \(\mathrm {ord}_\tau (x^{\prime })=\ell _\tau (x^{\prime })+1=\ell _\tau (x)+2\) . By Lemma REF  we have that \(\mathrm {ord}_\tau (x)\ge \mathrm {ord}_\tau (x^{\prime })-1\ge \ell _\tau (x)+1\) . By Lemma REF  we deduce that \(\ell _\tau (x)+1=\mathrm {ord}_\tau (x)\) . This completes the proof of the theorem.
Note that the above proof can be simplified in the case where \(W_{(\tau )}\)  is finite. In this case, denote by \(w_0\) , the maximal element of \(W_{(\tau )}\)  (for the Bruhat order \(\le _\tau \) ). Let \(x\in I_\tau (\tau ,\mathrm {gen},W_{(\tau )})\setminus \lbrace 0\rbrace \) . Let \(w\in \mathrm {supp}\big (\mathrm {LT}(x)\big )\) . Set \(y=\tilde{\mathcal {K}}_{w_0 w^{-1}}.x\) . Then \(\mathrm {supp}(\tilde{\mathcal {K}}_{w_0w^{-1}}.x)\ni w_0\) . By Lemma REF , \(\mathrm {ord}_\tau (\tilde{\mathcal {K}}_{w_{0}w^{-1}}.x)=\ell _\tau (w_0)+1\) . By Lemma REF , we deduce that
\(\mathrm {ord}_\tau (x)\ge \mathrm {ord}_\tau (y)-\ell (w_0w^{-1})=\ell _\tau (w_0)+1-\big (\ell (w_0)-\ell (w^{-1})\big )=\ell (w)+1.\) 
By Lemma REF , we deduce that \(\mathrm {ord}_\tau (x)=\ell _\tau (x)+1\) .
We obtain a version of the Knapp-Stein dimension theorem in our frameworks:
Corollary 4.7 (see [37]})
Let \(\tau \in \mathcal {U}_. Then \)\(I_\tau (\tau )=\bigoplus _{w_R\in R_\tau } F̏_{w_R}.\mathbf {v}_\tau .\)\( If moreover \) s=s'\( for all \) sS\(, then \) EndH-mod(I)R]\(.\)
This follows from [6]}.
Note that under the notation of [6]}, Theorem REF  implies that \(I_\tau (\tau ,W_{(\tau )})=_\tau \) . We can thus apply the results from 5.3 to 5.6 of [6]}, when \(\tau \in \mathcal {U}_. In particular, we have a description of the submodules and the irreducible quotients of \) I\( when \) U, see [6]}.
We also obtain Kato's irreducibility criterion:
Corollary 4.8 (see [42]})
Let \(\tau \in T_. Then \) I\( is irreducible if and only if:\begin{enumerate}\item \tau \in \mathcal {U}_,\end{enumerate}\item \) W()={1}\(.\)

By [1]}, if \(I_\tau \)  is irreducible, then \(W_\tau =W_{(\tau )}\)  and \(\tau \in \mathcal {U}_. Conversely, let \) U be such that \(W_{(\tau )}=W_\tau \) . Then \(R_\tau =W_\tau /W_{(\tau )}=\lbrace 1\rbrace \)  and by Corollary REF , \(I_\tau (\tau )=_\tau \) . By [1]} we deduce that \(I_\tau \)  is irreducible.
\(\)
\(\alpha _r,\alpha _r^\vee \)
\(\mathbb {A}\)
\(\mathcal {J}_\tau \)
\(\mathcal {S}\)
\(\Phi ,\Phi ^\vee \)
\(\sigma _s,\sigma _s^{\prime }\)
\(\sigma _{s,\tau }^{\prime \prime }\)
\(\tilde{\mathcal {K}}_s\)
\(\zeta _r\)
\(C^v_f\)
\(F_w\)
\(N_{\Phi ^\vee }(w)\)
\(T_\)
\(W^v\)
\({R}\)
\({S}\)
Université de Lorraine, CNRS, Institut Élie Cartan de Lorraine, F-54000 Nancy, France, UMR 7502
E-mail: auguste.hebert@univ-lorraine.fr.
